In this article, NLS rogue wave patterns are determined by the complex roots of the Yablonskii-Vorob'ev polynomial hierarchy, and these roots are the pole locations of rational solutions to the \(\mbox{P}_{\mbox{\scriptsize II}}\)  hierarchy (see Sec. REF  and [1]}, [2]}).
Interestingly, in very different contexts, somewhat similar results have also been reported. For instance, in the semiclassical NLS equation after wave breaking, a sequence of Peregrine waves appear, and their locations are determined by the poles of the tritronquée solution to the first Painlevé
(\(\mbox{P}_{\mbox{\scriptsize I}}\) ) equation [3]}. In the semiclassical sine-Gordon equation with initial conditions near the separatrix of a simple pendulum, superluminal (infinite velocity) kinks that appear in the solution are linked to the real roots of the Yablonskii-Vorob'ev polynomials associated with rational solutions of the \(\mbox{P}_{\mbox{\scriptsize II}}\)  equation [4]}. This connection of wave phenomena to rational solutions of the Painlevé equations may arise again in other wave systems in the future.
